Prove 1 + 2 + 3 + β¦β¦. + n = (π§(π§+π))/π for n, n is a natural number
Step 1: Let P(n) : (the given statement)
Let P(n): 1 + 2 + 3 + β¦β¦. + n = (n(n + 1))/2
Step 2: Prove for n = 1
For n = 1,
L.H.S = 1
R.H.S = (π(π + 1))/2 = (1(1 + 1))/2 = (1 Γ 2)/2 = 1
Since, L.H.S. = R.H.S
β΄ P(n) is true for n = 1
Step 3: Assume P(k) to be true and then prove P(k + 1) is true
Assume that P(k) is true,
P(k): 1 + 2 + 3 + β¦β¦. + k = (π(π + 1))/2
We will prove that P(k + 1) is true.
P(k + 1): 1 + 2 + 3 +β¦β¦. + (k + 1) = ((k + 1)( (k + 1) + 1))/2
P(k + 1): 1 + 2 + 3 +β¦β¦.+ k + (k + 1) = ((π€ + π)(π€ + π))/π
We have to prove P(k + 1) is true
Solving LHS
1 + 2 + 3 +β¦β¦.+ k + (k + 1)
From (1): 1 + 2 + 3 + β¦β¦. + k = (π(π + 1))/2
= (π(π + π))/π + (k + 1)
= (π(π + 1) + 2(π + 1))/2
= ((π + π)(π + π))/π
= RHS
β΄ P(k + 1) is true when P(k) is true
Step 4: Write the following line
Thus, By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.